Theorem nfcnv 5865

Description: Bound-variable hypothesis builder for converse relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

Ref	Expression
nfcnv.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcnv	⊢ Ⅎ𝑥◡𝐴

Proof of Theorem nfcnv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnv 5670	. 2 ⊢ ◡𝐴 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
2		nfcv 2931	. . . 4 ⊢ Ⅎ𝑥𝑧
3		nfcnv.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2931	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 5162	. . 3 ⊢ Ⅎ𝑥 𝑧𝐴𝑦
6	5	nfopab 5184	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
7	1, 6	nfcxfr 2929	1 ⊢ Ⅎ𝑥◡𝐴

Syntax hints:  Ⅎwnfc 2916   class class class wbr 5113  {copab 5177  ^◡ccnv 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-cnv 5670

This theorem is referenced by:  nfrn  5943  nfpred  6308  nffun  6560  nff1  6773  funcnvmpt  6992  nfsup  9410  nfinf  9442  gsumcom2  20044  ptbasfi  23706  mbfposr  25779  itg1climres  25841  nfwsuc  36206  aomclem8  43679  rfcnpre1  45630  rfcnpre2  45642  smfpimcc  47413